General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1915/16. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Gravitation is a natural Phenomenon by which objects with Mass attract one another Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical It unifies special relativity and Newton's law of universal gravitation, resulting in a description of gravity as a property of the geometry of space and time. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another In Physics, the treatment of Time is a central issue It has been treated as a question of Geometry. In particular, the curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of whatever matter and radiation are present. In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Matter is commonly defined as being anything that has mass and that takes up space. Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy The relation is specified by the Einstein field equations, a system of partial differential equations. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i

General relativity predicts novel effects relating to the passage of time, the geometry of space, the motion of bodies in free fall, and the propagation of light. Free fall is motion with no Acceleration other than that provided by Gravity. Light, or visible light, is Electromagnetic radiation of a Wavelength that is visible to the Human eye (about 400–700 Examples are gravitational time dilation, the gravitational redshift of light, and the gravitational time delay; the theory's predictions have been confirmed in numerous observations and experiments. Gravitational time dilation is the effect of time passing at different rates in regions of different Gravitational potential; the higher the local distortion of Spacetime In Physics, Light or other forms of Electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational The Shapiro time delay effect or gravitational time delay effect is one of the four classic solar system Tests of general relativity. At its introduction in 1915 the general theory of relativity did not have a solid empirical foundation Although not the only relativistic theory of gravity, general relativity is the simplest such theory that is consistent with the experimental data. Alternatives to general relativity are physical theories that attempt to describe the phenomena of Gravitation in competition to Einstein's theory of General Still, a number of open questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature

Einstein's theory has important astrophysical applications. It points towards the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not even light, can escape—as an end-state for massive stars. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A star is a massive luminous ball of plasma. The nearest star to Earth is the Sun, which is the source of most of the Energy on Earth There is evidence that, indeed, such stellar black holes as well as more massive varieties of black hole are responsible for the intense radiation emitted by certain types of astronomical objects (such as active galactic nuclei or microquasars). A stellar black hole is a Black hole formed by the Gravitational collapse of a massive Star (20 or more Solar masses, though the exact amount Radiation, as in Physics, is Energy in the form of waves or moving Subatomic particles emitted by an atom or other body as it changes from a higher energy An active galactic nucleus ( AGN) is a compact region at the centre of a Galaxy which has a much higher than normal luminosity over some or all of the Electromagnetic The bending of light by gravity can lead to the curious phenomenon of gravitational lensing, where multiple images of the same distant astronomical object are visible in the sky. A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of General relativity also predicts the existence of gravitational waves, which have since been measured indirectly; a direct measurement is the aim of projects such as LIGO. In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from For the Latvian holiday Ligo see Jāņi. LIGO stands for Laser Interferometer Gravitational-Wave Observatory. In addition, general relativity is the basis of current cosmological models of an expanding universe. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its

General relativity

Einstein field equations

Introduction to... Mathematical formulation of...

This box: view • talk • edit

History

Main articles: History of general relativity, Golden age of general relativity, and Classical theories of gravitation

Soon after publishing his theory of special relativity in 1905, Einstein started to think about how to incorporate gravity into his new relativistic framework. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the General relativity (GR is a Theory of Gravitation that was developed by Albert Einstein between 1907 and 1915 The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General Creation of General Relativity Early investigations As Albert Einstein later said the reason for the development of General relativity was that The Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of General relativity, which had previously been regarded as something of In Theoretical physics, the current Gold Standard Theory of Gravitation is the General theory of relativity. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Gravitation is a natural Phenomenon by which objects with Mass attract one another In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. A thought experiment (from the German Gedankenexperiment) is a proposal for an Experiment that would test a Hypothesis or Theory After numerous detours and false starts, his work culminated in the November, 1915 presentation to the Prussian Academy of Science of what are now known as the Einstein field equations. The Prussian Academy of Sciences (Preußische Akademie der Wissenschaften was an Academy established in Berlin on July 11 1700. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the These equations specify how the geometry of space and time is influenced by whatever matter is present, and form the core of Einstein's general theory of relativity. ^[1]

In his own derivation of his theory's predictions, Einstein had worked with approximation methods. But as early as 1916, the astrophysicist Schwarzschild found the first non-trivial exact solution to the Einstein field equations. Karl Schwarzschild ( October 9, 1873 - May 11, 1916) was a German Jewish Physicist and Astronomer. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside It is nowadays known under his name, and laid the groundwork for the description of the final stages of gravitational collapse, and the objects which eventually became known as a black holes. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e The same year saw the first steps of generalization to electrically charged objects that would result in the Reissner-Nordström solution, now associated with electrically charged black holes. In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational ^[2] In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its In line with contemporary thinking, he was set on describing a static universe and, to achieve that goal, added a new parameter to his original field equations, the cosmological constant. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification ^[3] When, in 1929, the work of Hubble and others made it clear that our universe is indeed expanding, and thus better described by expanding cosmological solutions found by Friedmann in 1922, Lemaître formulated the earliest version of the big bang models. Edwin Powell Hubble ( November 20, 1889 – September 28, 1953) was an American astronomer. Alexander Alexandrovich Friedman or Friedmann (Александр Александрович Фридман ( June 16 1888, Saint Petersburg, Imperial Georges Henri Joseph Édouard Lemaître ( July 17, 1894 &ndash June 20, 1966) was a Belgian Roman Catholic Priest The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. ^[4]

During all that time, general relativity remained something of a curiosity among physical theories. There was evidence that it was to be preferred to Newtonian gravity: Einstein himself had shown in 1915 how his theory effortlessly explained the anomalous perihelion advance of the planet Mercury,^[5] and a 1919 expedition led by Eddington had announced confirmation of general relativity's prediction for the deflection of starlight by the Sun^[6] (instantly catapulting Einstein to world fame^[7]). Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass At its introduction in 1915 the general theory of relativity did not have a solid empirical foundation Sir Arthur Stanley Eddington, OM (28 December 1882 – 22 November 1944 was an English Astrophysicist of the early 20th century Yet it was only with the developments between approximately 1960 and 1975, now known as the Golden age of general relativity, that the theory entered the mainstream of theoretical physics and astrophysics. The Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of General relativity, which had previously been regarded as something of Theoretical physics employs Mathematical models and Abstractions of Physics in an attempt to explain experimental data taken of the natural world Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, Physicists began to understand the concept of a black hole, and to identify these objects' astrophysical manifestation as quasars. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A quasar (contraction of QUASi-stellAR radio source) is an extremely powerful and distant Active galactic nucleus. ^[8] Ever more precise solar system tests confirmed the theory's predictive power,^[9] and relativistic cosmology, too, became amenable to direct observational tests. ^[10]

From classical mechanics to general relativity

General relativity is best understood by examining its similarities with and departures from classical physics. The first step is the realization that classical mechanics and Newton's law of gravity admit of a geometric description. Combining this description with the laws of special relativity, one can heuristically derive general relativity. ^[11]

Geometry of Newtonian gravity

At the base of classical mechanics is the notion that a body's motion can be described as a combination of free (or inertial) motion, and deviations from this free motion. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving Such deviations are caused by external forces acting on a body in accordance with Newton's second law of motion, which states that the force acting on a body is equal to that body's (inertial) mass times its acceleration. In Physics, a force is whatever can cause an object with Mass to Accelerate. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the In Physics, a force is whatever can cause an object with Mass to Accelerate. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object ^[12] The preferred inertial motions are related to the geometry of space and time: in the standard reference frames of classical mechanics, objects in free motion move along straight lines at constant speed. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of In modern parlance, their paths are geodesics, straight world lines in spacetime. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS ^[13]

Conversely, one might expect that inertial motions, once identified by observing the actual motions of bodies and making allowances for the external forces (such as electromagnetism or friction), can be used to define the geometry of space, as well as a time coordinate. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point However, there is an ambiguity once gravity comes into play. Gravitation is a natural Phenomenon by which objects with Mass attract one another According to Newton's law of gravity, and independently verified by experiments such as that of Eötvös and its successors, there is a universality of free fall (also known as the weak equivalence principle, or the universal equality of inertial and passive-gravitational mass): the trajectory of a test body in free fall depends only on its position and initial speed, but not on any of its material properties. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass Baron Loránd von Eötvös, more commonly called Baron Roland von Eötvös in the English literature (in Hungarian Vásárosnaményi Báró Eötvös Loránd, or The equivalence principle The equivalence principle In physical theories, a test particle is an idealized model of an object whose physical properties (usually Mass, Charge, or size) are assumed Free fall is motion with no Acceleration other than that provided by Gravity. ^[14] A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard an accelerated rocket. ^[15]

Given the universality of free fall, there is no observable distinction between inertial motion and motion under the influence of the gravitational force. This suggests the definition of a new class of inertial motion, namely that of objects in free fall under the influence of gravity. This new class of preferred motions, too, defines a geometry of space and time—in mathematical terms, it is the geodesic motion associated with a specific connection which depends on the gradient of the gravitational potential. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces Potential energy can be thought of as Energy stored within a physical system Space, in this construction, still has the ordinary Euclidean geometry. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. However, as can be shown using simple thought experiments following the free-fall trajectories of different test particles, the result of transporting spacetime vectors that can denote a particle's velocity (time-like vectors) will vary with the particle's trajectory; in mathematical terms: the Newtonian connection is not integrable. In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems. From this, one can deduce that spacetime is curved. In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object The result is a geometric formulation of Newtonian gravity using only covariant concepts, in other words: a description which is valid in any desired coordinate system. ^[16] In this geometric description, tidal effects—the relative acceleration of bodies in free fall—are related to the derivative of the connection, showing how the modified geometry is caused by the presence of mass. The tidal force is a secondary effect of the Force of Gravity and is responsible for the Tides It arises because the gravitational acceleration experienced ^[17]

Relativistic generalization

As intriguing as geometric Newtonian gravity may be, its basis, classical mechanics, is merely a limiting case of (special) relativistic mechanics. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial ^[18] In the language of symmetry: where gravity can be neglected, physics is Lorentz invariant as in special relativity rather than Galilei invariant as in classical mechanics. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or In standard Physics, Lorentz covariance is a key property of Spacetime that follows from the Special theory of relativity, where it applies globally Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial (The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations. In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime ) The differences between the two become significant when we are dealing with speeds approaching the speed of light, and with high-energy phenomena. ^[19]

With Lorentz symmetry, additional structures comes into play. They are defined by the set of light cones (see the image on the left). The light-cones define a causal structure: for each event A, there is a set of events that can, in principle, either influence or be influenced by A via signals or interactions that do not need to travel faster than light (such as event B in the image), and a set of events for which such an influence is impossible (such as event C in the image). These sets are observer-independent. ^[20] But the light-cones, in conjunction with the world-lines of freely falling particles, contain even more information: they can be used to reconstruct the space-time's semi-Riemannian metric, at least up to a positive scalar factor. In mathematical terms, this defines a conformal structure. In Mathematics, conformal geometry is the study of the set of angle-preserving ( conformal) transformations on a Riemannian manifold or Pseudo-Riemannian ^[21]

Special relativity is defined in the absence of gravity, so for practical applications, it is a suitable model whenever gravity can be neglected. Bringing gravity into play, and assuming the universality of free fall, an analogous reasoning as in the previous section applies: there are no global inertial frames. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest Instead there are approximate inertial frames moving alongside freely falling particles. Translated into the language of spacetime: the straight time-like lines that define a gravity-free inertial frame are deformed to lines that are curved relative to each other, suggesting that the inclusion of gravity necessitates a change in spacetime geometry. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS ^[22]

A priori, it is not clear whether the new local frames in free fall are indeed those in which the laws of special relativity hold—that theory is based on the propagation of light, and thus on electromagnetism, which could have a different set of preferred frames. Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of But using different assumptions about the special-relativistic frames (such as their being earth-fixed, or in free fall), one can derive different predictions for the gravitational redshift, that is, the way in which the frequency of light shifts as the light propagates through a gravitational field (cf. In Physics, Light or other forms of Electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational below). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 The actual measurements show that free-falling frames are the ones in which light propagates as it does in special relativity. ^[23] The generalization of this statement, namely that the laws of special relativity hold, to good approximation, in freely falling (and non-rotating) reference frames, is known as the Einstein equivalence principle, a crucial guiding principle for generalizing special-relativistic physics to include gravity. The equivalence principle ^[24]

The same experimental data shows that time as measured by clocks in a gravitational field—proper time, to give the technical term—does not follow the rules of special relativity. In relativity, proper time is Time measured by a single Clock between events that occur at the same place as the clock In the language of spacetime geometry, it is not measured by the Minkowski metric. In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity As in the Newtonian case, this is suggestive of a more general geometry. At small scales, all reference frames that are in free fall are equivalent, and approximately Minkowskian. Consequently, we are now dealing with a curved generalization of Minkowski space. The metric tensor that defines the geometry—in particular, how lengths and angles are measured—is not the Minkowski metric of special relativity, it is a generalization known as a semi- or pseudo-Riemannian metric. This article discusses metrics in General relativity, for a discussion of metrics in general see Metric tensor. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. Furthermore, each Riemannian metric is naturally associated with one particular kind of connection, the Levi-Civita connection, and this is, in fact, the connection that satisfies the equivalence principle and makes space locally Minkowskian (that is, in suitable "locally inertial" coordinates, the metric is Minkowskian, and its derivatives and the connection coefficients vanish). In Riemannian geometry, the Levi-Civita connection is the torsion -free Riemannian connection, i ^[25]

Einstein's equations

Main article: Einstein field equations

Having formulated the relativistic, geometric version of the effects of gravity, the question of gravity's source remains. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Newtonian gravity, the source is mass. In special relativity, mass turns out to be part of a more general quantity called the energy-momentum tensor, which includes both energy and momentum densities as well as stress (that is, pressure and shear). The stress-energy tensor (sometimes stress-energy-momentum tensor is a Tensor quantity in Physics that describes the Density and Flux Energy density is the amount of Energy stored in a given system or region of space per unit Volume, or per unit Mass, depending on the context although In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different Stress is a measure of the average amount of Force exerted per unit Area. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface ^[26] Using the equivalence principle, this tensor is readily generalized to curved space-time. Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely. A field equation is an equation in a Physical theory that describes how a Fundamental force (or a combination of such forces interacts with Matter In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In special relativity, conservation of energy-momentum correspond to the statement that the energy-momentum tensor is divergence-free. Energy conservation is the practice of decreasing the quantity of energy used In Vector calculus, the divergence is an Operator that measures the magnitude of a Vector field &rsquos source or sink at a given point the This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. With this additional condition—the covariant divergence of the energy-momentum tensor, and hence of whatever is on the other side of the equation, is zero— the simplest set of equations are what are called Einstein's (field) equations. They equate the energy-momentum tensor and a specific divergence-free combination of the Ricci tensor and the metric, which is known as the Einstein tensor:

where G_ab is the Einstein tensor, T_ab is the energy-momentum tensor (both written in abstract index notation). In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined The Einstein tensor expresses Spacetime curvature in the Einstein field equations for Gravitation in the Theory of general relativity. Abstract index notation is a mathematical notation for Tensors and Spinors that uses indices to indicate their types rather than their components in a particular basis ^[27] Matching the theory's prediction to observational results for planetary orbits (or, equivalently, assuring that the weak-gravity, low-speed limit is Newtonian mechanics), the proportionality constant can be fixed as κ = 8πG/c⁴, with G the gravitational constant and c the speed of light. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass ^[28]

It is worth mentioning that there are alternatives to general relativity built upon the same premises, which include additional rules and/or constraints, leading to different field equations. Alternatives to general relativity are physical theories that attempt to describe the phenomena of Gravitation in competition to Einstein's theory of General Examples are Brans-Dicke theory, teleparallelism, and Einstein-Cartan theory. In Theoretical physics, the Brans-Dicke theory of gravitation (sometimes called the Jordan-Brans-Dicke theory) is a theoretical framework to explain Gravitation Teleparallelism (also called distant parallelism and teleparallel gravity) was an attempt by Einstein to unify Electromagnetism and Gravity Einstein–Cartan theory in Theoretical physics extends General relativity to correctly handle Spin angular momentum. ^[29]

Definition and basic applications

See also: Mathematics of general relativity and Physical theories modified by general relativity

The derivation outlined in the previous section contains all the information needed to define and characterize general relativity. The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying Albert Einstein 's theory of General This article will use the Einstein summation convention. The theory of General relativity required the adaptation of existing theories of physical electromagnetic Having the defined the theory, we will address a question of crucial importance in physics: how the theory can be used for model-building.

Definition and basic properties

General relativity is a metric theory of gravitation. This article discusses metrics in General relativity, for a discussion of metrics in general see Metric tensor. Gravitation is a natural Phenomenon by which objects with Mass attract one another At its core are Einstein's equations, which describe the relation between the geometry of a four-dimensional, semi-Riemannian manifold representing spacetime on the one hand, and the energy-momentum contained in that spacetime on the other. The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. ^[30] Phenomena that, in classical mechanics, are ascribed to the action of the force of gravity (such as free-fall, orbital motion, and spacecraft trajectories), correspond to inertial motion within a curved geometry of spacetime in general relativity: there is no gravitational force deflecting objects from their natural, straight paths. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Free fall is motion with no Acceleration other than that provided by Gravity. In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star A spacecraft is a Vehicle or machine designed for Spaceflight. Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry Instead, gravity changes the properties of space and time, which changes the straightest-possible paths that objects will naturally follow. ^[31] The curvature is, in turn, caused by the energy-momentum of matter. Paraphrasing the relativist John Archibald Wheeler, spacetime tells matter how to move; matter tells spacetime how to curve. John Archibald Wheeler ( July 9, 1911 &ndash April 13, 2008) was an eminent American Theoretical physicist. ^[32]

While general relativity replaces the scalar gravitational potential of classical physics by a symmetric rank-two tensor, the latter reduces to the former in certain limiting cases. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually This article discusses quantum theory For other uses see Correspondence principle (disambiguation. For weak gravitational fields and slow speed relative to the speed of light, the theory's predictions converge on those of Newton's law of gravity. Linearized gravity is an approximation scheme in General relativity in which the nonlinear contributions from the Spacetime metric are ignored In physics the slow-motion approximation is an approximation that is used in relativistic mechanics where the Speed of an object under consideration is significantly Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass ^[33]

As it is constructed using tensors, general relativity exhibits general covariance: its laws—and further laws formulated within the general relativistic framework—take on the same form in all coordinate systems. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually In Theoretical physics, general covariance (also known as Diffeomorphism covariance or general invariance) is the Invariance of the In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point ^[34] Furthermore, the theory does not contain any invariant geometric background structures. It thus satisfies a more stringent general principle of relativity, namely that the laws of physics are the same for all observers. A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in A physical law or scientific law is a Scientific generalization based on empirical Observations of physical behavior (i ^[35] Locally, as expressed in the equivalence principle, spacetime is Minkowskian, and the laws of physics have local Lorentz invariance. The equivalence principle In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity A local Lorentz covariance or local Lorentz symmetry is a Local symmetry of Space-time which reduces locally (i ^[36]

Model-building

The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Where appropriate this article will use the Abstract index notation. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, and the matter must satisfy whatever additional equations have been imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present. ^[37]

Einstein's equations are non-linear partial differential equations and, as such, very difficult to solve. This article describes the use of the term nonlinearity in mathematics In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i ^[38] Nevertheless, a number of exact solutions are known, although only a few of them have direct physical applications. In General relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter ^[39] The best-known exact solutions, and also those most interesting from a physics point of view, are the Schwarzschild solution, the Reissner-Nordström solution and the Kerr metric, each corresponding to a certain type of black hole in an otherwise empty universe,^[40] and the Friedmann-Lemaître-Robertson-Walker and de Sitter universes, each describing an expanding cosmos. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In Physics and Astronomy, the Reissner-Nordström metric is a solution to the Einstein field equations in empty space which corresponds to the gravitational In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A de Sitter universe is a solution to Einstein 's field equations of General Relativity which is named after Willem de Sitter. ^[41] Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and Anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture). The Gödel metric is an exact solution of the Einstein field equations in which the Stress-energy tensor contains two terms the first representing the This article details time travel itself For other uses see Time Traveler. The Taub-NUT vacuum is an exact solution to Einstein's equations, a model Universe formulated in the framework of General relativity that is Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity In Mathematics and Physics, n -dimensional anti de Sitter space, sometimes written AdS_n is a maximally symmetric Lorentzian manifold For the relation of the AdS/CFT correspondence to the general context of string theory see String theory. ^[42]

Significant efforts are being made in the field of numerical relativity, where powerful computers are employed to find interesting numerical solutions describing, say, two black holes orbiting each other. Numerical relativity is one of the branches of General relativity that uses numerical methods and algorithms to solve and analyze problems A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e ^[43] Also, there are different methods for finding approximate solutions in the context of perturbation theory. This article describes perturbation theory as a general mathematical method The best-known of these are linearized gravity^[44] and its generalization, the Post-Newtonian expansion. Linearized gravity is an approximation scheme in General relativity in which the nonlinear contributions from the Spacetime metric are ignored Post-Newtonian expansions in General relativity are used for finding an approximate solution of the Einstein equations for the Metric tensor that The latter provides for a systematic way of describing a spacetime that contains matter which is not particularly compact, and which moves slowly compared with the speed of light. The expansion involves a series of terms. The first terms represent Newtonian gravity. Additional terms systematically represent smaller and smaller effects arising from the difference between Newton's theory and general relativity. ^[45] An extension of this expansion is the Parametrized Post-Newtonian (PPN) formalism, a framework for making quantitative comparisons between the predictions of general relativity and alternative theories. Post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear equations of gravity in terms of the lowest-order deviations from Newton's theory Alternatives to general relativity are physical theories that attempt to describe the phenomena of Gravitation in competition to Einstein's theory of General ^[46]

Consequences of Einstein's theory

General relativity has a number of physical consequence. Some follow directly from the theory's axioms, while others have become clear only in the course of the ninety years of research that followed Einstein's initial publication.

Gravitational time dilation and frequency shift

Main article: Gravitational time dilation

In any theory in which the equivalence principle holds,^[47] gravity has an immediate influence on the passage of time. Gravitational time dilation is the effect of time passing at different rates in regions of different Gravitational potential; the higher the local distortion of Spacetime Light sent down into a gravity well is blueshifted, light climbing out of a gravity well, redshifted in what is known as the gravitational frequency shift. In Physics and Astronomy, redshift occurs when Electromagnetic radiation – usually Visible light – emitted or reflected by Closely related is the fact that processes near massive bodies run more slowly when compared with processes taking place further away; an effect known as gravitational time dilation. Gravitational time dilation is the effect of time passing at different rates in regions of different Gravitational potential; the higher the local distortion of Spacetime ^[48]

Gravitational redshift has been measured in the laboratory^[49] and using astronomical observations. ^[50] Gravitational time dilation in the Earth's gravitational field has been measured numerous times using atomic clocks,^[51] while ongoing validation is provided as a side-effect of the operation of the Global Positioning System (GPS). An atomic clock is a type of Clock that uses an Atomic resonance Frequency standard as its timekeeping element Basic concept of GPS operation A GPS receiver calculates its position by carefully timing the signals sent by the constellation of GPS Satellites high above the Earth ^[52] Tests in stronger gravitational fields are provided by the observation of binary pulsars. A binary pulsar is a Pulsar with a binary companion, often another Pulsar, White dwarf or Neutron star. ^[53] All results are in agreement with general relativity. ^[54] However, at the current level of accuracy, these observations cannot distinguish between general relativity and other theories in which the equivalence principle is valid. ^[55]

Light deflection and gravitational time delay

Main articles: Kepler problem in general relativity, Gravitational lens, and Shapiro delay

In general relativity, light follows a special variety of straightest-possible world-line, so-called light-like or null geodesics—a generalization of the straight lines along which light travels in classical physics, and the invariance of lightspeed in special relativity. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces Invariance is a French journal edited by Jacques Camatte, published since 1968. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial ^[56] As one examines suitable model spacetimes (either the exterior Schwarzschild solution or, for more than a single mass, the Post-Newtonian expansion),^[57] several effects of gravity on light propagation emerge. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Post-Newtonian expansions in General relativity are used for finding an approximate solution of the Einstein equations for the Metric tensor that

The best-known is the bending of light in a gravitational field: light passing a massive body is deflected towards that body. While such an effect can also be derived by extending the universality of free fall to light,^[58] the maximal angle of deflection resulting from such heuristic calculations is only half the value given by general relativity. The equivalence principle ^[59] Important observable examples involve the light of stars or distant quasars being deflected as it passes the Sun. A quasar (contraction of QUASi-stellAR radio source) is an extremely powerful and distant Active galactic nucleus. The Sun (Sol is the Star at the center of the Solar System. ^[60]

Closely related to light deflection is the gravitational time delay, also known as the Shapiro effect: light signals take longer to move through a gravitational field than they would in the absence of the gravitational field. The Shapiro time delay effect or gravitational time delay effect is one of the four classic solar system Tests of general relativity. There have been numerous successful tests of this prediction. ^[61] In the parameterized post-Newtonian formalism (PPN), measurements of both the deflection of light and the gravitational time delay are used to determine a parameter called γ that reflects the influence of gravity on the geometry of space. Post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear equations of gravity in terms of the lowest-order deviations from Newton's theory ^[62]

Gravitational waves

Main article: Gravitational waves

There are several analogies between weak-field gravity and electromagnetism. In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of One is that, for electromagnetic waves, there are corresponding gravitational waves: spacetime ripples which propagate at the speed of light. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from ^[63]

The simplest variety of gravitational wave can be visualized via their action on a ring of freely floating particles (see first image to the right). As a simple sine wave propagates through such a ring from out of the page towards the reader, the ring is distorted in a characteristic, rhythmic fashion (see second image to the right). ^[64] Such linearized gravitational waves are important when it comes to describing the exceedingly weak waves that are expected to arrive here on Earth from far-off cosmic events, which typically result in distances increasing and decreasing by 10 ^{− 21} or less. Data analysis methods routinely make use of the fact that these linearized waves can be Fourier decomposed. In Mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions ^[65]

Due to the non-linearity of the Einstein equations, there is no linear superposition for arbitrarily strong gravitational waves, making their description a difficult task. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems There are some exact solutions describing gravitational waves, for instance a wave train traveling through empty space^[66] or so-called Gowdy universes, varieties of an expanding cosmos filled with gravitational waves. In mathematics and especially Physics, an exact solution is a Solution to a problem that encapsulates the whole mathematics or physics of the problem without using Gowdy universes or alternatively Gowdy solutions of Einstein's equations are simple model Spacetimes in General relativity which represent an ^[67] On the other hand, when it comes to a sufficiently complete description of the gravitational waves produced in astrophysically relevant situations such as the merger of two black holes, numerical methods are presently the only way to construct appropriate models. Numerical relativity is one of the branches of General relativity that uses numerical methods and algorithms to solve and analyze problems ^[68]

Orbital effects and the relativity of direction

Main article: Kepler problem in general relativity

General relativity differs from classical mechanics in a number of predictions concerning orbiting bodies. The Kepler problem in general relativity involves solving for the motion of two spherical bodies interacting with one another by Gravitation, as described by the theory of It predicts the relativistic apside shifts, as well as orbital decay caused by the emission of gravitational waves and effects which are due to the relativity of direction. In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from

Precession of apsides

In general relativity, the apsides of orbits (the points of an orbiting body's closest approach to the system's center of mass) will precess—the orbit is not an ellipse, but akin to an ellipse that rotates on its focus, resulting in a rosette-like shape (see image). In Celestial mechanics, an apsis, plural apsides (ˈæpsɨdɪːz is the point of greatest or least distance of the Elliptical orbit of an object from In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star Precession refers to a change in the direction of the axis of a rotating object In Mathematics, an ellipse (from the Greek ἔλλειψις literally absence) is a Conic section, the locus of points in a Einstein himself derived this result by using an approximate metric representing the Newtonian limit and treating the orbiting body like a test particle. In physical theories, a test particle is an idealized model of an object whose physical properties (usually Mass, Charge, or size) are assumed For him, the fact that his theory gave a straightforward explanation of the anomalous perihelion shift of the planet Mercury, discovered earlier by Urbain Le Verrier in 1859, was important evidence that he had at last identified the correct form of the gravitational field equations. At its introduction in 1915 the general theory of relativity did not have a solid empirical foundation Urbain Jean Joseph Le Verrier ( March 11, 1811 &ndash September 23, 1877) was a French Mathematician who specialized in Celestial The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the ^[69]

The effect can also be derived by using either the exact Schwarzschild metric (describing spacetime around a spherical mass)^[70] or the much more general post-Newtonian formalism. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside Post-Newtonian expansions in General relativity are used for finding an approximate solution of the Einstein equations for the Metric tensor that ^[71] It is due both to the influence of gravity on the geometry of space and to the way that self energy contributes to a body's gravity (one effect of the special kind of nonlinearity exhibited by Einstein's theory). In Theoretical physics and Quantum field theory a particle's self-energy \Sigma represents the contribution to the particle's Energy, or This article describes the use of the term nonlinearity in mathematics ^[72] Relativistic precession has been observed for all planets that allow for accurate precession measurements (Mercury, Venus and the Earth),^[73] as well as in binary pulsar systems, where it is larger by five orders of magnitude. The VENUS ( V ictoria E xperimental N etwork U nder the S ea project is a cabled sea floor observatory operated by the University EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 A binary pulsar is a Pulsar with a binary companion, often another Pulsar, White dwarf or Neutron star. An order of magnitude is the class of scale or magnitude of any amount where each class contains values of a fixed ratio to the class preceding it ^[74]

Orbital decay

According to general relativity, a binary system will emit gravitational waves, thereby losing energy. A binary system is an astronomical term referring to two objects in space (usually Stars but also Planets galaxies or Asteroids which In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός Due to this loss, the distance between the two orbiting bodies decreases, and so does their orbital period. Within the solar system or for ordinary double stars, the effect is too small to be observable. The Solar System consists of the Sun and those celestial objects bound to it by Gravity. Double Star is a Science fiction Novel by Robert A Heinlein, first serialized in Astounding Science Fiction Not so for a close binary pulsar, a system of two orbiting neutron stars, one of which is a pulsar: from the pulsar, observers on Earth receive a regular series of radio pulses that can serve as a highly accurate clock, which allows precise measurements of the orbital period. A binary pulsar is a Pulsar with a binary companion, often another Pulsar, White dwarf or Neutron star. A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type Pulsars are highly magnetized rotating Neutron stars that emit a beam of Electromagnetic radiation in the form of radio waves Since the neutron stars are very compact, significant amounts of energy are emitted in the form of gravitational radiation. ^[76]

The first observation of a decrease in orbital period due to the emission of gravitational waves was made by Hulse and Taylor using the binary pulsar PSR1913+16 that they had discovered in 1974. Russell Alan Hulse (born November 28, 1950) is an American Physicist and winner of the Nobel Prize in Physics, shared with his Joseph Hooton Taylor Jr (born March 29, 1941) is an American Astrophysicist and Nobel Prize in Physics laureate It amounts to the first indirect detection of gravitational waves, rewarded with the Nobel Prize in physics in 1993. The Nobel Prize (Nobelpriset (Nobelprisen is a Swedish prize established in the 1895 will of Swedish chemist Alfred Nobel; it was first awarded in Peace, Literature ^[77] Since then, several other binary pulsars have been found, the most spectacular find being the double pulsar PSR J0737-3039 in which both stars are pulsars. |- style="vertical-align top"| Distance | 1600 - 2000 Ly (600 Parsecs) PSR J0737-3039 is a binary Pulsar system ^[78]

Geodetic precession and frame-dragging

Main articles: Geodetic precession and Frame dragging

Several relativistic effects are directly related to the relativity of direction. The geodetic effect represents the effect of the curvature of Spacetime, predicted by General relativity, on a spinning moving body Albert Einstein 's theory of General relativity predicts that rotating bodies drag Spacetime around themselves in a phenomenon referred to as frame-dragging ^[79] One is geodetic precession: for a gyroscope in free fall in curved spacetime, the direction of its axis will change when compared, for instance, with the direction of light received from distant stars—even though its motion comes closest to keeping its axis direction constant ("parallel transport"). The geodetic effect represents the effect of the curvature of Spacetime, predicted by General relativity, on a spinning moving body A gyroscope is a device for measuring or maintaining orientation, based on the principles of Angular momentum. In Geometry, parallel transport is a way of transporting geometrical data along smooth curves in a Manifold. ^[80] For the Moon-Earth-system, this effect has been measured with the help of lunar laser ranging;^[81] more recently, it has been measured for test masses aboard the satellite Gravity Probe B to a precision of better than 1 percent. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 The ongoing Lunar Laser Ranging Experiment measures the distance between the Earth and the Moon using laser ranging. Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004 ^[82]

Near a rotating mass, there are so-called gravitomagnetic or frame-dragging effects: for a distant observer, it will seem that objects close to the mass get "dragged around". Albert Einstein 's theory of General relativity predicts that rotating bodies drag Spacetime around themselves in a phenomenon referred to as frame-dragging This is most extreme for rotating black holes where, for an object entering a zone known as the ergosphere, rotation is inevitable. In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body The ergosphere is a region located outside a Rotating black hole. ^[83] Such effects can again be tested through their influence on the orientation of a gyroscope in free fall. ^[84] Somewhat controversial tests have been performed using the LAGEOS satellites, confirming the relativistic prediction. LAGEOS, or Laser Geodynamics Satellites are a series of scientific research Satellites designed to provide an orbiting laser ranging benchmark for geodynamical ^[85] A precision measurement is the main aim of the Gravity Probe B mission, whose final results are expected in September 2008. Gravity Probe B ( GP-B) is a Satellite -based mission which launched in 2004 ^[86]

Astrophysical applications

Gravitational lensing

Main article: Gravitational lensing

The deflection of light by gravity can lead to an intriguing phenomenon. A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of The Einstein Cross or Q2237+030 or QSO 2237+0305 is a gravitationally lensed Quasar that sits directly behind ZW 2237+030 Huchra's A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of If there is a massive object between the observer and a distant target object, with the mass and relative distances just right, the observer will see multiple distorted images of the target. This and similar effects are known as gravitational lensing^[87] and, depending on the configuration, scale, and mass distribution, it can result in two or more images, a bright ring known as an Einstein ring, or partial rings called arcs. A gravitational lens is formed when the light from a very distant bright source (such as a Quasar) is "bent" around a massive object (such as a cluster of In observational Astronomy an Einstein ring is the deformation of the light from a source (such as a Galaxy or Star) into a ring through Gravitational ^[88] The earliest example was discovered in 1979;^[89] since then, more than a hundred gravitational lenses have been observed. The Twin Quasar (Double Quasar or Old Faithful is also known as Q0957+561, or QSO 0957+561. ^[90] Images too close to be resolved can still lead to a measurable effect, namely an overall brightening of a given star or other point-like object; a number of such "microlensing events" has been observed, as well. Gravitational microlensing is an astronomical phenomenon due to the Gravitational lens effect ^[91]

Gravitational lensing has developed into a tool of observational astronomy. Observational astronomy is a division of the astronomical Science that is concerned with getting data in contrast with Theoretical astrophysics which is Notably, it is used to detect the presence and distribution of dark matter, provide a "natural telescope" for observing distant galaxies, and obtain an independent estimate of the Hubble constant. In Physics and cosmology, dark matter is hypothetical Matter that does not interact with the electromagnetic force but whose presence can be inferred from Hubble's law is the statement in Physical cosmology that the Redshift in light coming from distant galaxies is proportional to their distance Statistical evaluations of lensing data are also used to understand the structural evolution of galaxies. A galaxy is a massive gravitationally bound system consisting of Stars an Interstellar medium of gas and dust, and Dark matter ^[92]

Gravitational wave astronomy

Main article: Gravitational waves

Observations of binary pulsars provide strong indirect evidence for the existence of gravitational waves (see the section on Orbital decay, above). In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from The Laser Interferometer Space Antenna (LISA experiment is a joint venture of NASA and the European Space Agency (ESA to detect and observe in detail Gravitational General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 However, gravitational waves reaching us from the depths of the cosmos have not been detected directly, this being one of the major goals of current relativity-related research. ^[93] To this end, a number of land-based gravitational wave detectors are currently in operation, most notably the interferometric detectors GEO 600, LIGO (three detectors), TAMA 300 and VIRGO. A gravitational wave detector is any experiment designed to measure Gravitational waves, minute distortions of Spacetime that are predicted by Einsteins An interferometric gravitational wave detector is a Gravitational wave detector that uses Laser Interferometry to detect the influence of Gravitational GEO 600 is a Gravitational wave detector located in Hannover, Germany For the Latvian holiday Ligo see Jāņi. LIGO stands for Laser Interferometer Gravitational-Wave Observatory. TAMA 300 is a Gravitational wave detector located at the Mitaka campus of the National Astronomical Observatory of Japan. The Virgo detector for Gravitational waves consists mainly in a Michelson Laser Interferometer made of two orthogonal arms being each 3 kilometers ^[94] A joint US-European mission to launch a space-based detector, LISA, is currently under development,^[95] with a precursor mission (LISA Pathfinder) due for launch in late 2009. The Laser Interferometer Space Antenna (LISA experiment is a joint venture of NASA and the European Space Agency (ESA to detect and observe in detail Gravitational LISA Pathfinder is the revised name for SMART-2 an ESA space probe to be launched in 2009. ^[96]

Gravitational waves promise to yield information about astronomical objects that is inaccessible by observations using electromagnetic radiation:^[97] Terrestrial detectors are expected to yield new information about inspiral phase and mergers of binary stellar mass black holes and binaries consisting of one such black hole and a neutron star (of interest as a candidate mechanism for gamma ray bursts). Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. A stellar black hole is a Black hole formed by the Gravitational collapse of a massive Star (20 or more Solar masses, though the exact amount A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type Gamma-ray bursts ( GRB s are the most luminous electromagnetic events occurring in the Universe since the Big Bang. They could also detect signals from core-collapse supernovae and from periodic sources such as rotating neutron stars with small deformation. Type II Supernova, or core-collapse supernova, is a sub-category of cataclysmic Variable stars that results from the internal collapse and violent explosion If there is truth to speculation about certain kinds of phase transitions or kink bursts from long cosmic strings in the very early universe (at cosmic times around 10 ^{− 25} seconds) these could also be detectable. In Thermodynamics, phase transition or phase change is the transformation of a thermodynamic system from one phase to another A cosmic string is a hypothetical 1-dimensional (spatially Topological defect in various fields Cosmic time (also known as "time since the big bang" is the Time Coordinate commonly used in the Big Bang models of Physical cosmology ^[98] Space-based detectors like LISA should detect objects such as binaries consisting of two White Dwarfs, and AM CVn stars (a White Dwarf accreting matter from its binary partner, a low-mass helium star), and also observe the mergers of supermassive black holes and the inspiral of smaller objects (between one and a thousand solar masses) into such black holes. A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. A white dwarf, also called a degenerate dwarf, is a small Star composed mostly of Electron-degenerate matter. A supermassive black hole is a Black hole with a Mass of an order of magnitude between 105 and 1 The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies. LISA should also be able to listen to the same kind of sources from the early universe as ground-based detectors, but at even lower frequencies and with greatly increased sensitivity. ^[99]

Black holes and other compact objects

Main article: Black hole

Whenever an object becomes sufficiently compact, general relativity predicts the formation of a black hole: a region of space from which nothing, not even light, can escape. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e In the currently accepted models of stellar evolution, neutron stars with around 1. Stellar evolution is the process by which a Star undergoes a sequence of radical changes during its lifetime A neutron star is a type of remnant that can result from the Gravitational collapse of a massive Star during a Type II, Type Ib or Type 4 solar mass and so-called stellar black holes with a few to a few dozen solar masses are thought to be the final state for the evolution of massive stars. The solar mass is a standard way to express Mass in Astronomy, used to describe the masses of other Stars and galaxies. A stellar black hole is a Black hole formed by the Gravitational collapse of a massive Star (20 or more Solar masses, though the exact amount ^[100] Supermassive black holes with a few million to a few billion solar masses are considered the rule rather than the exception in the centers of galaxies,^[101] and their presence is thought to have played an important role in the formation of galaxies and larger cosmic structures. A supermassive black hole is a Black hole with a Mass of an order of magnitude between 105 and 1 ^[102]

Astronomically, the most important property of compact objects is that they provide a superbly efficient mechanism for converting gravitational energy into electromagnetic radiation. ^[103] Accretion, the falling of dust or gaseous matter onto stellar or supermassive black holes, is thought to be responsible for some spectacularly luminous astronomical objects, notably diverse kinds of active galactic nuclei on galactic scales and stellar-size objects such as Microquasars. Dust is a general name for minute Solid particles with Diameters less than 500 micrometers. A stellar black hole is a Black hole formed by the Gravitational collapse of a massive Star (20 or more Solar masses, though the exact amount A supermassive black hole is a Black hole with a Mass of an order of magnitude between 105 and 1 An active galactic nucleus ( AGN) is a compact region at the centre of a Galaxy which has a much higher than normal luminosity over some or all of the Electromagnetic ^[104] In particular, accretion can lead to relativistic jets, focused beams of highly energetic particles that are being flung into space at almost light speed. The lower-energy non-relativistic version of this phenomenon is described at Polar jet. ^[105] Interestingly, to a distant observer, some of these jets even appear to move faster than light; this, however, can be explained as an optical illusion that does not violate the tenets of relativity. In Astronomy, superluminal motion is the apparently Faster-than-light motion seen in some radio galaxies, Quasars and recently also in some galactic ^[106] General relativity plays a central role in modelling all these phenomena,^[107] relativistic lensing effects being thought to play a role for the signals received from X-ray pulsars. X-ray pulsars or accretion-powered pulsars are a class of astronomical objects that are X-ray sources displaying strict periodic variations in X-ray intensity ^[108]

Limits on compactness from the observation of accretion-driven phenomena ("Eddington luminosity"),^[109] observations of stellar dynamics in the center of our own Milky Way galaxy,^[110] and indications that at least some of the compact objects in question appear to have no solid surface^[111] provide strong indirect evidence for the existence of black holes. The Eddington luminosity (also referred to as the Eddington limit) in a star is defined as the point where the gravitational force inwards equals the continuum radiation force The Milky Way (a translation of the Latin Via Lactea, in turn derived from the Greek Γαλαξίας (Galaxias sometimes referred to simply Direct evidence, such as observing the "shadow" of the Milky Way galaxy's central black hole horizon, is eagerly sought for. ^[112]

Black holes are also sought-after targets in the search for gravitational waves (see the section Gravitational waves, above): merging black hole binaries should lead to some of the strongest gravitational wave signals reaching detectors here on Earth, and reliable simulations of such mergers are one of the main goals of current research in numerical relativity. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 Numerical relativity is one of the branches of General relativity that uses numerical methods and algorithms to solve and analyze problems ^[113] The phase directly before the merger ("chirp") could be used as a "standard candle" to deduce the distance to the merger events, and hence as a probe of cosmic expansion at large distances. A standard candle is an astronomical object that has a known Luminosity. ^[114] The gravitational waves produced as a stellar black hole plunges into a supermassive one should serve as a probe of the supermassive black hole's geometry. ^[115]

Cosmology

Main article: Physical cosmology

Each solution of Einstein's equations describes a whole universe, so it should come as no surprise that there are solutions that provide useful models for cosmology, the study of the universe as a whole. Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The current models are based on an extension of the original form of Einstein's equations which include the cosmological constant Λ, an additional term that has an important influence on the large-scale dynamics of the cosmos,

where g_ab is the spacetime metric. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification This article discusses metrics in General relativity, for a discussion of metrics in general see Metric tensor. ^[116]

On the basis of isotropic and homogeneous solutions of these enhanced equations, the so-called Friedmann-Lemaître-Robertson-Walker solutions,^[117] are built the models of modern cosmology in which the universe has evolved over the past 14 billion years from a hot, early Big Bang phase. Isotropy is uniformity in all directions Precise definitions depend on the subject area Physical cosmology, as a branch of Astronomy, is the study of the large-scale structure of the Universe and is concerned with fundamental questions about its The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. ^[118] Once a small number of parameters (for example the universe's mean matter density) have been fixed by astronomical observation,^[119] further observational data can be used to put the models to the test: successful predictions include the initial abundance of chemical elements formed in a period of primordial nucleosynthesis,^[120] which is in good agreement with astronomical observations,^[121] and the large-scale distribution of galaxies. Matter is commonly defined as being anything that has mass and that takes up space. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In Physical cosmology, Big Bang nucleosynthesis (or primordial nucleosynthesis) refers to the production of nuclei other than those of H-1 (i ^[122] Further important evidence is provided by the existence and properties of a "thermal echo" from the early cosmos, the cosmic background radiation, which contains information about the cosmological parameters, the universes matter and energy content, and the seeds of today's large scale structure. A thermal column (or thermal) is a column of rising Air in the lower altitudes of the Earth's atmosphere. ^[123] Future measurements could also reveal evidence about gravitational waves in the early universe. ^[124]

The status of the resulting models is mixed. On the one hand, the standard models of cosmology have been very successful: to date, they have passed all observational tests,^[125] and they have proven a sound basis to explaining the evolution of the universe's large-scale structure. ^[126] On the other hand, there are a number of important open questions. The determination of cosmological parameters (in line with other astronomical observations^[127]) suggests that about 90 percent of all matter in the universe is in the form of so-called dark matter, which has mass (and hence gravitational influence), but does not interact electromagnetically (and hence cannot be observed directly). In Physics and cosmology, dark matter is hypothetical Matter that does not interact with the electromagnetic force but whose presence can be inferred from There is currently no generally accepted description of this new kind of matter within the framework of particle physics^[128] or otherwise. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them ^[129] A similar open question is that of dark energy. In Physical cosmology, dark energy is a hypothetical exotic form of Energy that permeates all of space and tends to increase the rate of expansion of the universe Observational evidence from redshift surveys of distant supernovae and measurements of the cosmic background radiation show that the evolution of our universe is significantly influenced by a cosmological constant resulting in an acceleration of cosmic expansion or, equivalently, by a form of energy with an unusual equation of state, namely dark energy. A supernova (plural supernovae or supernovas) is a stellar Explosion. In Physical cosmology, the cosmological constant (usually denoted by the Greek capital letter Lambda: Λ was proposed by Albert Einstein as a modification In Physics and Thermodynamics, an equation of state is a relation between state variables More specifically an equation of state is a thermodynamic ^[130] The nature of this new form of energy remains unclear. ^[131]

A number of further problems of the classical cosmological models (such as "why is the cosmic background radiation so highly homogeneous")^[132] have led to the introduction of an additional phase of strongly accelerated expansion at cosmic times of around 10 ^{− 33} seconds, known as an inflationary phase. In Physical cosmology, cosmic inflation is the idea that the nascent Universe passed through a phase of exponential expansion that ^[133] While recent measurements of the cosmic background radiation have resulted in first evidence for this scenario,^[134] problems remain. There is a bewildering variety of possible inflationary scenarios not restricted by current observations. ^[135] Also, the question remains what happened in the earliest universe, close to where the classical models predict the big bang singularity. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become An authoritative answer would require a complete theory of quantum gravity, which does not exist at the moment^[136] (cf. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature the section on quantum gravity, below). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

Advanced concepts

Causal structure and global geometry

Main article: Causal structure

In general relativity, no material body can catch up with or over take a light pulse; no influence from an event A can reach any other location before light sent out at A does so. The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold Hence, an exploration of all light worldlines (null geodesics) yields key information about the spacetime's causal structure. In General relativity, Geodesics generalize the notion of "straight lines" to curved Spacetime. This structure can be displayed using Penrose-Carter diagrams in which infinitely large regions of space and infinite time intervals are shrunk ("compactified") so as to fit onto a finite map, while light still travels along diagonals as in standard spacetime diagrams. In Theoretical physics, a Penrose diagram (named for mathematical physicist Roger Penrose) is a Two-dimensional Diagram that captures the In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact. The Minkowski diagram was developed in 1908 by Herman Minkowski and provides an illustration of the properties of space and time in the Special theory of relativity ^[137]

Aware of the importance of causal structure, Roger Penrose and others developed important techniques that are now termed global geometry. Sir Roger Penrose, PhD, OM, FRS (born 8 August 1931) is an English Mathematical physicist and Emeritus Spacetime topology, the topological structure of Spacetime, is a subject studied primarily in General relativity. In global geometry, the object of study is not one particular solution (or family of solutions) to Einstein's equations. Where appropriate this article will use the Abstract index notation. Rather, relations that hold true for all geodesics, such as the Raychaudhuri equation, are utilized in conjunction with non-specific assumptions about the nature of matter (usually in the form of so-called energy conditions) to derive general results. In General relativity, Raychaudhuri's equation is a fundamental result describing the motion of nearby bits of matter Matter is commonly defined as being anything that has mass and that takes up space. In relativistic classical field theories of Gravitation, particularly General relativity, an energy condition is one of various alternative conditions ^[138]

Horizons

Main articles: Horizon (general relativity), No hair theorem, and Black hole mechanics

One of the most striking conclusions that can be drawn from studies of global geometry is the existence of boundaries called horizons, which demarcate one spacetime region from the rest of the spacetime. There are several types of Spacetime horizons that play a role in Einstein 's theory of General relativity: Absolute horizon, a boundary The no-hair theorem in Astrophysics postulates that all Black hole solutions of the Einstein-Maxwell equations of Gravitation and Electromagnetism In Physics, black hole thermodynamics is the area of study that seeks to reconcile the Laws of thermodynamics with the existence of Black hole Event In General relativity, an event horizon is a boundary in Spacetime, an area surrounding a Black hole or a Wormhole, inside which events cannot SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS The best-known examples are black holes: if mass is compressed into a sufficiently compact region of space (as specified in the hoop conjecture, the relevant length scale being the Schwarzschild radius^[139]), spacetime is partitioned into an inside and an outside world: no light from the inside can escape to the outside, and since, in general relativity, no object can overtake a light pulse, all inside matter is imprisoned as well. A black hole is a theoretical region of space in which the Gravitational field is so powerful that nothing not even Electromagnetic radiation (e The Hoop Conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a Black hole when and only when a circular hoop The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a characteristic Radius associated with every Mass. However, matter and radiation may cross the horizon into the black hole—clearly showing that horizons are not physical barriers. The resulting object is known as a black hole, and the surface in question as the black hole's horizon. ^[140]

Initial black hole studies relied on simplified models obtained from explicit solutions of Einstein's equation, notably the spherically-symmetric Schwarzschild solution (used to describe a static black hole) and the axisymmetric Kerr solution (used to describe a rotating, stationary black hole, and introducing interesting features such as the ergosphere). The ergosphere is a region located outside a Rotating black hole. In mathematics and especially Physics, an exact solution is a Solution to a problem that encapsulates the whole mathematics or physics of the problem without using The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In General relativity, a Spacetime is said to be static if it admits a global nowhere zero Timelike Hypersurface orthogonal Killing In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body In General relativity, a Spacetime is said to be stationary if it admits a global nowhere zero Timelike Killing vector field. The ergosphere is a region located outside a Rotating black hole. Subsequent studies using global geometry have revealed more general properties of black holes: in the long run, they are rather simple objects characterized by eleven parameters specifying energy, linear momentum, angular momentum, location at a specified time and electric charge. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. This is stated by the black hole uniqueness theorems: "black holes have no hair", that is, no distinguishing marks like the hairstyles of humans: irrespective of the complexity of a gravitating object collapsing to form a black hole, the object that results (having emitted gravitational waves) is very simple. The no-hair theorem in Astrophysics postulates that all Black hole solutions of the Einstein-Maxwell equations of Gravitation and Electromagnetism In Physics, a gravitational wave is a Fluctuation in the Curvature of Spacetime which propagates as a wave, traveling outward from ^[141]

Even more remarkably, there is a general set of laws known as black hole mechanics, which is analogous to the laws of thermodynamics. In Physics, black hole thermodynamics is the area of study that seeks to reconcile the Laws of thermodynamics with the existence of Black hole Event The laws of thermodynamics, in principle describe the specifics for the transport of Heat and work in Thermodynamic processes. For example, by the second law of black hole mechanics, the area of the event horizon of a general black hole will never decrease with time, analogous to the entropy of a thermodynamic system. In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy This limits the energy that can be extracted from a rotating black hole (e. g. by the Penrose process). The Penrose process (also called Penrose mechanism) is a process theorised by Roger Penrose wherein energy can be extracted from a Rotating black hole. ^[142] There is strong evidence that the laws of black hole mechanics are, in fact, a special case of the laws of thermodynamics, and that the black hole area does indeed denote its entropy:^[143] semi-classical calculations indicate that black holes do emit thermal radiation, with the surface gravity playing the role of temperature in Planck's law. Thermal radiation is Electromagnetic radiation emitted from the surface of an object which is due to the object's Temperature. For a general introduction see Black body. In Physics, Planck's law describes the spectral radiance of Electromagnetic radiation This radiation is known as Hawking radiation (cf. Hawking radiation (also known as Bekenstein-Hawking radiation) is a Thermal radiation with a black body spectrum predicted to be emitted by Black holes the quantum theory section, below). General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 ^[144]

Horizons also play a role for other kinds of solutions. In an expanding universe, some regions of the past can be unobservable ("particle horizon"), and some regions of the future cannot be influenced (event horizon). In Physical cosmology, particle horizon is the maximum distance from which particles could have traveled to the observer in the Age of the universe In both cases, the location of the horizon in spacetime depends on the event in question. ^[145] Even in flat Minkowski space, when described by an accelerated observer (Rindler space), there will be horizons^[146] (associated with a semi-classical radiation known as Unruh radiation). In Relativistic physics, the Rindler coordinate chart is an important and useful Coordinate chart representing part of flat Spacetime, also called the The Unruh effect, discovered in 1976 by Bill Unruh of the University of British Columbia, is the prediction that an accelerating observer will observe ^[147]

Singularities

Main article: Spacetime singularity

Another general—and quite disturbing—feature of general relativity is the appearance of spacetime boundaries known as singularities. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become Ordinary spacetime can be explored by following up on all possible ways that light and particles in free fall can travel (that is, all timelike and lightlike geodesics). But there are spacetimes which fulfill all the requirements of Einstein's theory, yet have "ragged edges"—regions where the paths of light and falling particles come to an abrupt end and geometry becomes ill-defined. By definition, these are spacetime singularities. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become In more interesting cases, the geometrical quantities characterizing spacetime curvature (e. g. the Ricci scalar) take on infinite values at such "curvature singularities". In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest Curvature invariant of a Riemannian manifold. ^[148] Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which describes a singularity inside an eternal static black hole,^[149] or the Kerr solution with its ring-shaped singularity inside an eternal rotating black hole. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. In Einstein's theory of General relativity, the Schwarzschild solution (or the Schwarzschild vacuum) describes the Gravitational field outside In General relativity, the Kerr metric (or Kerr vacuum) describes the geometry of Spacetime around a rotating massive body ^[150] The Friedmann-Lemaître-Robertson-Walker solutions, and other spacetimes describing universes, have past singularities on which worldlines begin, namely big bang singularities, and some have future singularities (big crunch) as well. The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation. In Physical cosmology, the Big Crunch is one possible scenario for the Ultimate fate of the universe, in which the Metric expansion of space eventually ^[151]

From these examples, all highly symmetric and thus simplified, one might think the occurrence of singularities to be a consequence of idealization. The famous singularity theorems proved using the methods of global geometry say otherwise: singularities are a generic feature of general relativity, and unavoidable once the collapse of an object with realistic matter properties has proceeded beyond a certain stage^[152] and also at the beginning of a wide class of expanding universes. The Penrose-Hawking singularity theorems are a set of results in General relativity which attempt to answer the question of whether gravity is necessarily singular ^[153] However, these theorems say very little about the properties of singularities, and much of current research is devoted to characterizing these entities' generic structure (hypothesized e. g. by the so-called BKL conjecture). ^[154] The cosmic censorship hypothesis states that all realistic future singularities (no perfect symmetries, matter with realistic properties) are safely hidden away behind a horizon, and thus invisible to all distant observers. The weak and the strong Cosmic Censorship Hypotheses are two mathematical conjectures about the structure of singularities arising in General relativity. While no formal proof yet exists, numerical simulations offer supporting evidence of its validity. ^[155]

Evolution equations

Main article: Initial value formulation (general relativity)

Each solution of Einstein's equation encompasses the whole history of a universe—it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. The initial value formulation is a way of expressing the formalism of Einstein's theory of General relativity in a way that describes a Universe evolving Where appropriate this article will use the Abstract index notation. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems: if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems) For Einstein's equations, there appear to be subtle differences compared with other fields, for example, they are self-interacting (that is, non-linear even in the absence of other fields, and they have no fixed background structure—the stage itself evolves as the cosmic drama is played out). The Einstein field equations ( EFE) or Einstein's equations are a set of ten equations in Einstein 's theory of General relativity in which the In Physics, a field is a Physical quantity associated to each point of Spacetime. This article describes the use of the term nonlinearity in mathematics ^[156]

Nevertheless, in order to understand Einstein's equations as partial differential equations, it is crucial to re-formulate them in a way that describes the evolution of the universe over time. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i This is achieved by so-called "3+1" formulations, where spacetime is split into three space dimensions and one time dimension, such as the ADM formalism. The ADM Formalism developed by Arnowitt, Deser and Misner is a Hamiltonian formulation of General relativity. ^[157] These decompositions show that the spacetime evolution equations of general relativity are indeed well-behaved, meaning that solutions always exist and are uniquely defined (once suitable initial conditions are specified). In Mathematics, an existence theorem is a theorem with a statement beginning 'there exist(s. The method of image charges (also known as the method of images and method of mirror charges) is a basic problem-solving tool in Electrostatics. ^[158] Formulations like this are also the basis of numerical relativity: attempts to simulate the evolution of relativistic spacetimes (notably merging black holes or gravitational collapse) using computers. Numerical relativity is one of the branches of General relativity that uses numerical methods and algorithms to solve and analyze problems ^[159]

Global and quasi-local quantities

Main article: Mass in general relativity

The notion of evolution equations is intimately tied in with another aspect of general relativistic physics. The concept of Mass in General relativity (GR is more complex than the concept of Mass in special relativity. In Einstein's theory, it turns out to be impossible to find a general definition for a seemingly simple property such as a system's total mass (or energy). Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The main reason for this is that the gravitational field—like any physical field—must be ascribed a certain energy. However, it is fundamentally impossible to localize that energy. ^[160]

Nevertheless, there are possibilities to define a system's total mass, either using a hypothetical "infinitely distant observer" (ADM mass)^[161] or suitable symmetries (Komar mass). In Theoretical physics, the ADM energy (short for Richard '''A'''rnowitt, Stanley '''D'''eser and Charles '''M'''isner) is a special way to define Introduction and motivation The Komar mass of a system is one of several formal concepts of Mass that used in General relativity. ^[162] If one excludes from the system's total mass the energy being carried away to infinity by gravitational waves, the result is the so-called Bondi mass at null infinity. The concept of Mass in General relativity (GR is more complex than the concept of Mass in special relativity. ^[163] Just as in classical physics, it can be shown that these masses are positive. ^[164] Analogous global definitions exist for momentum and angular momentum. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position ^[165] In addition, there have been a number of attempts to define quasi-local quantities, such as the mass of an isolated system formulated using only quantities defined within a finite region of space containing that system. The hope is to obtain a quantity useful for general statements about isolated systems, such as a more precise formulation of the hoop conjecture. In the Natural sciences an isolated system, as contrasted with a open system, is a Physical system that does not interact with its Surroundings The Hoop Conjecture, proposed by Kip Thorne in 1972, states that an imploding object forms a Black hole when and only when a circular hoop ^[166]

Relationship with quantum theory

Along with general relativity, quantum theory, the basis of our understanding of matter from elementary particles to solid state physics, is considered one of the two pillars of modern physics. In Particle physics, an elementary particle or fundamental particle is a particle not known to have substructure that is it is not known to be made Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and ^[167] However, it is still an open question of how the concepts of quantum theory can be reconciled with those of general relativity.

Quantum field theory in curved spacetime

Main article: Quantum field theory in curved spacetime

Ordinary quantum field theories, which form the basis of modern elementary particle physics, are defined in flat Minkowski space, which is an excellent approximation when it comes to describing the behavior of microscopic particles in weak gravitational fields like those found on Earth. Quantum field theory in curved spacetime is an extension of standard Quantum field theory to curved spacetime. In quantum field theory (QFT the forces between particles are mediated by other particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Physics and Mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of Special relativity ^[168] In order to describe situations in which gravity is strong enough to influence (quantum) matter, yet not strong enough to require quantization itself, physicists have formulated quantum field theories in curved spacetime. These theories rely on classical general relativity to describe a curved background spacetime, and define a generalized quantum field theory to describe the behavior of quantum matter within that spacetime. ^[169] Using this formalism, it can be shown that black holes emit a blackbody spectrum of particles known as Hawking radiation, leading to the possibility that the evaporate over time. Hawking radiation (also known as Bekenstein-Hawking radiation) is a Thermal radiation with a black body spectrum predicted to be emitted by Black holes Hawking radiation (also known as Bekenstein-Hawking radiation) is a Thermal radiation with a black body spectrum predicted to be emitted by Black holes ^[170] As briefly mentioned above, this radiation plays an important role for the thermodynamics of black holes. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 ^[171]

Quantum gravity

Main article: Quantum gravity

See also: String theory and Loop quantum gravity

The demand for consistency between a quantum description of matter and a geometric description of spacetime,^[172] as well as the appearance of singularities (where curvature length scales become microscopic), indicate the need for a full theory of quantum gravity: for an adequate description of the interior of black holes, and of the very early universe, a theory is required in which gravity and the associated geometry of spacetime are described in the language of quantum physics. Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings Loop quantum gravity (LQG, also known as loop gravity and Quantum geometry, is a proposed quantum theory of Spacetime which attempts to reconcile the theories A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature ^[173] Despite major efforts, no complete and consistent theory of quantum gravity is currently known, even though a number of promising candidates exists. ^[174]

Attempts to generalize ordinary quantum field theories, used in elementary particle physics to describe fundamental interactions, so as to include gravity have led to serious problems. In mathematics Calabi&ndashYau manifolds are compact Kähler manifolds whose Canonical bundle is trivial Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them At low energies, this approach proves successful, in that it results in an acceptable effective (quantum) field theory of gravity. In Physics, an effective field theory is an approximate theory (usually a Quantum field theory) that includes appropriate degrees of freedom to describe ^[175] At very high energies, however, the result are models devoid of all predictive power ("non-renormalizability"). In Quantum field theory, the Statistical mechanics of fields and the theory of self-similar geometric structures renormalization refers to a collection ^[176]

One attempt to overcome these limitations is to formulate a quantum theory not of point particles, but of minute one-dimensional extended objects: string theory. In Physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in Quantum physics A point particle (or point-like, often spelled pointlike) is an idealized object heavily used in Physics. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings ^[177] The theory promises to be a unified description of all particles and interactions, including gravity;^[178] the price to pay are unusual features such as six extra dimensions of space in addition to the usual three. A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and ^[179] In what is called the second superstring revolution, it was conjectured that both string theory and a unification of general relativity and supersymmetry known as supergravity^[180] form part of a hypothesized eleven-dimensional model known as M-theory, which would constitute a uniquely defined and consistent theory of quantum gravity. The second superstring revolution was the intense wave of breakthroughs in String theory that took place approximately between 1994 and 1997. In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that In Theoretical physics, supergravity ( supergravity theory) is a field theory that combines the principles of Supersymmetry and General relativity In Theoretical physics, M-theory is a new limit of String theory in which 11 dimensions of Spacetime may be identified ^[181]

Another approach starts with the canonical quantization procedures of quantum theory. In Physics, canonical quantization is one of many procedures for quantizing a Classical theory. Using the initial-value-formulation of general relativity (cf. the section on evolution equations, above), the result is an analogue of the Schrödinger equation: the Wheeler-deWitt equation which, regrettably, turns out to be ill-defined. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system ^[182] However, with the introduction of what are now known as Ashtekar variables,^[183] this leads to a promising model known as Loop quantum gravity. In Theoretical physics, Ashtekar (new variables (named after Abhay Ashtekar who invented them represent an unusual way to rewrite the metric on the three-dimensional Loop quantum gravity (LQG, also known as loop gravity and Quantum geometry, is a proposed quantum theory of Spacetime which attempts to reconcile the theories Space is represented by a network structure called a spin network, evolving over time in discrete steps. In Physics, a spin network is a type of diagram which can be used to represent states and interactions between particles and fields in Quantum physics ^[184]

Depending on which features of general relativity and quantum theory are accepted unchanged, and on what level changes are introduced,^[185] there are numerous other attempts to arrive at a viable theory of quantum gravity, some example being dynamical triangulations,^[186] causal sets,^[187] twistor models^[188] or the path-integral based models of quantum cosmology. The twistor theory, originally developed by Roger Penrose in 1967, is the mathematical theory which maps the Geometric objects of the four dimensional space-time In Theoretical physics, Quantum cosmology is a field attempting to study the effect of Quantum mechanics on the creation of the universe or ^[189]

All candidate theories still have major formal and conceptual problems to overcome. They also face the common problem that, as yet, there is no way to put quantum gravity predictions to experimental tests, although there is hope for this to change as future data from cosmological observations and particle physics experiments becomes available. ^[190]

Current status

General relativity has emerged as a highly successful model of gravitation and cosmology, which has so far passed every unambiguous observational and experimental test. Even so, there are strong indications the theory is incomplete. ^[191] The problem of quantum gravity and the question of the reality of spacetime singularities remain open. A gravitational singularity (sometimes spacetime singularity) is approximately a place where quantities which are used to measure the Gravitational field become ^[192] Observational data that is taken as evidence for dark energy and dark matter could indicate the need for new physics,^[193] and while the so-called Pioneer anomaly might yet admit of a conventional explanation, it, too, could be a harbinger of new physics. The Pioneer anomaly or Pioneer effect is the observed deviation from expectations of the trajectories of various Unmanned spacecraft visiting the Outer ^[194] Even taken as is, general relativity is rich with possibilities for further exploration. Mathematical relativists seek to understand the nature of singularities and the fundamental properties of Einstein's equations,^[195] and increasingly powerful computer simulations (such as those describing merging black holes) are run. ^[196] The race for the first direct detection of gravitational waves continues apace,^[197] in the hope of creating opportunities to test the theory beyond the limited approximations it has been tested so far even in the binary pulsar measurements. A binary pulsar is a Pulsar with a binary companion, often another Pulsar, White dwarf or Neutron star. ^[198] More than ninety years after its publication, general relativity remains a highly active area of research. ^[199]

See also

• Contributors to general relativity
• Einstein-Hilbert action
• General relativity resources, an annotated reading list giving bibliographic information on some of the most cited resources
• Introduction to mathematics of general relativity

Notes

References

External links

• Yale University Video Lecture: Special and General Relativity at Google Video
• Relativity: The special and general theory PDF
• Video Lectures on General Relativity by MIT Physics Professor Edmund Bertschinger.
• Series of lectures on General Relativity given in 2006 at the Institut Henri Poincaré (introductory courses and advanced ones).
• General Relativity Tutorials by John Baez

John Carlos Baez (born 1961 is an American mathematical physicist at the University of California Riverside.

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic